Homework 3

Details

For each problem make sure you clearly and neatly show all of your work! Each problem is worth 5 points.

1. Express π in 32-bit floating-point. To be more precise, express the closest number to π that you can with 32-bit floating-point notation.
   (The point of this question is to help you understand how floating-point notation works.)
2. How much error is there in the 32-bit representation of π?
   (The purpose of this question is to help you understand the limitations of floating-point notation.)
3. What number is the closest to 0 (not including 0) that you can store with 32-bit floating point? Express it both in scientific decimal notation and the 32-bit floating point notation. Explain clearly why it is the closest to zero that you can store.
   (The purpose of this question is to help you understand completely what the 3 parts of the floating-point representation do and what the limitations of each are.)

1. Remember that the smallest exponent you can legally store is -126, not -127 as you might expect.
2. Remember that the 1 before the decimal point is implied and not actually stored!
3. We did not specify this before, but if the mantissa is all 0s, that is the representation of 0 (so the 1 is not appended to the front if the whole mantissa is 0s). Given that, what is the smallest value you can represent in the mantissa?